TSTP Solution File: LCL520+1 by Twee---2.4.2
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : LCL520+1 : TPTP v8.1.2. Released v3.3.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n003.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 08:19:18 EDT 2023
% Result : Theorem 0.19s 0.63s
% Output : Proof 0.19s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : LCL520+1 : TPTP v8.1.2. Released v3.3.0.
% 0.00/0.13 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.12/0.33 % Computer : n003.cluster.edu
% 0.12/0.33 % Model : x86_64 x86_64
% 0.12/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.33 % Memory : 8042.1875MB
% 0.12/0.33 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.33 % CPULimit : 300
% 0.12/0.33 % WCLimit : 300
% 0.12/0.33 % DateTime : Fri Aug 25 06:09:08 EDT 2023
% 0.12/0.34 % CPUTime :
% 0.19/0.63 Command-line arguments: --set-join --lhs-weight 1 --no-flatten-goal --complete-subsets --goal-heuristic
% 0.19/0.63
% 0.19/0.63 % SZS status Theorem
% 0.19/0.63
% 0.19/0.67 % SZS output start Proof
% 0.19/0.67 Take the following subset of the input axioms:
% 0.19/0.67 fof(and_1, axiom, and_1 <=> ![X, Y]: is_a_theorem(implies(and(X, Y), X))).
% 0.19/0.67 fof(kn1, axiom, kn1 <=> ![P]: is_a_theorem(implies(P, and(P, P)))).
% 0.19/0.67 fof(kn2, axiom, kn2 <=> ![Q, P2]: is_a_theorem(implies(and(P2, Q), P2))).
% 0.19/0.67 fof(kn3, axiom, kn3 <=> ![R, P2, Q2]: is_a_theorem(implies(implies(P2, Q2), implies(not(and(Q2, R)), not(and(R, P2)))))).
% 0.19/0.67 fof(modus_ponens, axiom, modus_ponens <=> ![X2, Y2]: ((is_a_theorem(X2) & is_a_theorem(implies(X2, Y2))) => is_a_theorem(Y2))).
% 0.19/0.67 fof(op_and, axiom, op_and => ![X2, Y2]: and(X2, Y2)=not(or(not(X2), not(Y2)))).
% 0.19/0.67 fof(op_implies_and, axiom, op_implies_and => ![X2, Y2]: implies(X2, Y2)=not(and(X2, not(Y2)))).
% 0.19/0.67 fof(op_implies_or, axiom, op_implies_or => ![X2, Y2]: implies(X2, Y2)=or(not(X2), Y2)).
% 0.19/0.67 fof(op_or, axiom, op_or => ![X2, Y2]: or(X2, Y2)=not(and(not(X2), not(Y2)))).
% 0.19/0.67 fof(principia_op_and, axiom, op_and).
% 0.19/0.67 fof(principia_op_implies_or, axiom, op_implies_or).
% 0.19/0.67 fof(principia_r3, conjecture, r3).
% 0.19/0.67 fof(r3, axiom, r3 <=> ![P2, Q2]: is_a_theorem(implies(or(P2, Q2), or(Q2, P2)))).
% 0.19/0.67 fof(rosser_kn1, axiom, kn1).
% 0.19/0.67 fof(rosser_kn2, axiom, kn2).
% 0.19/0.67 fof(rosser_kn3, axiom, kn3).
% 0.19/0.67 fof(rosser_modus_ponens, axiom, modus_ponens).
% 0.19/0.67 fof(rosser_op_implies_and, axiom, op_implies_and).
% 0.19/0.67 fof(rosser_op_or, axiom, op_or).
% 0.19/0.67
% 0.19/0.67 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.19/0.67 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.19/0.67 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.19/0.67 fresh(y, y, x1...xn) = u
% 0.19/0.67 C => fresh(s, t, x1...xn) = v
% 0.19/0.67 where fresh is a fresh function symbol and x1..xn are the free
% 0.19/0.67 variables of u and v.
% 0.19/0.67 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.19/0.67 input problem has no model of domain size 1).
% 0.19/0.67
% 0.19/0.67 The encoding turns the above axioms into the following unit equations and goals:
% 0.19/0.67
% 0.19/0.67 Axiom 1 (rosser_op_or): op_or = true.
% 0.19/0.67 Axiom 2 (principia_op_and): op_and = true.
% 0.19/0.67 Axiom 3 (rosser_op_implies_and): op_implies_and = true.
% 0.19/0.67 Axiom 4 (principia_op_implies_or): op_implies_or = true.
% 0.19/0.67 Axiom 5 (rosser_modus_ponens): modus_ponens = true.
% 0.19/0.67 Axiom 6 (rosser_kn1): kn1 = true.
% 0.19/0.67 Axiom 7 (rosser_kn2): kn2 = true.
% 0.19/0.67 Axiom 8 (rosser_kn3): kn3 = true.
% 0.19/0.67 Axiom 9 (and_1): fresh57(X, X) = true.
% 0.19/0.67 Axiom 10 (r3): fresh9(X, X) = true.
% 0.19/0.67 Axiom 11 (modus_ponens_2): fresh60(X, X, Y) = true.
% 0.19/0.67 Axiom 12 (kn1_1): fresh33(X, X, Y) = true.
% 0.19/0.67 Axiom 13 (modus_ponens_2): fresh28(X, X, Y) = is_a_theorem(Y).
% 0.19/0.67 Axiom 14 (modus_ponens_2): fresh59(X, X, Y, Z) = fresh60(modus_ponens, true, Z).
% 0.19/0.67 Axiom 15 (kn2_1): fresh31(X, X, Y, Z) = true.
% 0.19/0.67 Axiom 16 (op_and): fresh24(X, X, Y, Z) = and(Y, Z).
% 0.19/0.67 Axiom 17 (op_implies_and): fresh22(X, X, Y, Z) = implies(Y, Z).
% 0.19/0.67 Axiom 18 (op_implies_or): fresh21(X, X, Y, Z) = implies(Y, Z).
% 0.19/0.67 Axiom 19 (op_implies_or): fresh21(op_implies_or, true, X, Y) = or(not(X), Y).
% 0.19/0.67 Axiom 20 (op_or): fresh20(X, X, Y, Z) = or(Y, Z).
% 0.19/0.67 Axiom 21 (op_implies_and): fresh22(op_implies_and, true, X, Y) = not(and(X, not(Y))).
% 0.19/0.67 Axiom 22 (kn3_1): fresh29(X, X, Y, Z, W) = true.
% 0.19/0.67 Axiom 23 (op_or): fresh20(op_or, true, X, Y) = not(and(not(X), not(Y))).
% 0.19/0.67 Axiom 24 (op_and): fresh24(op_and, true, X, Y) = not(or(not(X), not(Y))).
% 0.19/0.67 Axiom 25 (kn1_1): fresh33(kn1, true, X) = is_a_theorem(implies(X, and(X, X))).
% 0.19/0.67 Axiom 26 (and_1_1): fresh58(and_1, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 0.19/0.67 Axiom 27 (kn2_1): fresh31(kn2, true, X, Y) = is_a_theorem(implies(and(X, Y), X)).
% 0.19/0.67 Axiom 28 (modus_ponens_2): fresh59(is_a_theorem(implies(X, Y)), true, X, Y) = fresh28(is_a_theorem(X), true, Y).
% 0.19/0.67 Axiom 29 (and_1): fresh57(is_a_theorem(implies(and(x9, y9), x9)), true) = and_1.
% 0.19/0.67 Axiom 30 (r3_1): fresh8(r3, true, X, Y) = is_a_theorem(implies(or(X, Y), or(Y, X))).
% 0.19/0.67 Axiom 31 (r3): fresh9(is_a_theorem(implies(or(p3, q3), or(q3, p3))), true) = r3.
% 0.19/0.67 Axiom 32 (kn3_1): fresh29(kn3, true, X, Y, Z) = is_a_theorem(implies(implies(X, Y), implies(not(and(Y, Z)), not(and(Z, X))))).
% 0.19/0.67
% 0.19/0.67 Lemma 33: is_a_theorem(implies(and(X, Y), X)) = fresh58(and_1, op_or, X, Y).
% 0.19/0.67 Proof:
% 0.19/0.67 is_a_theorem(implies(and(X, Y), X))
% 0.19/0.67 = { by axiom 26 (and_1_1) R->L }
% 0.19/0.67 fresh58(and_1, true, X, Y)
% 0.19/0.67 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.67 fresh58(and_1, op_or, X, Y)
% 0.19/0.67
% 0.19/0.67 Lemma 34: fresh58(and_1, op_or, X, Y) = op_or.
% 0.19/0.67 Proof:
% 0.19/0.67 fresh58(and_1, op_or, X, Y)
% 0.19/0.67 = { by lemma 33 R->L }
% 0.19/0.67 is_a_theorem(implies(and(X, Y), X))
% 0.19/0.67 = { by axiom 27 (kn2_1) R->L }
% 0.19/0.67 fresh31(kn2, true, X, Y)
% 0.19/0.67 = { by axiom 7 (rosser_kn2) }
% 0.19/0.67 fresh31(true, true, X, Y)
% 0.19/0.67 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.67 fresh31(op_or, true, X, Y)
% 0.19/0.67 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.67 fresh31(op_or, op_or, X, Y)
% 0.19/0.67 = { by axiom 15 (kn2_1) }
% 0.19/0.67 true
% 0.19/0.67 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.67 op_or
% 0.19/0.67
% 0.19/0.67 Lemma 35: op_or = and_1.
% 0.19/0.67 Proof:
% 0.19/0.67 op_or
% 0.19/0.67 = { by axiom 1 (rosser_op_or) }
% 0.19/0.67 true
% 0.19/0.67 = { by axiom 9 (and_1) R->L }
% 0.19/0.67 fresh57(op_or, op_or)
% 0.19/0.67 = { by axiom 1 (rosser_op_or) }
% 0.19/0.67 fresh57(op_or, true)
% 0.19/0.67 = { by lemma 34 R->L }
% 0.19/0.67 fresh57(fresh58(and_1, op_or, x9, y9), true)
% 0.19/0.67 = { by lemma 33 R->L }
% 0.19/0.67 fresh57(is_a_theorem(implies(and(x9, y9), x9)), true)
% 0.19/0.67 = { by axiom 29 (and_1) }
% 0.19/0.67 and_1
% 0.19/0.67
% 0.19/0.67 Lemma 36: not(and(X, not(Y))) = implies(X, Y).
% 0.19/0.67 Proof:
% 0.19/0.67 not(and(X, not(Y)))
% 0.19/0.67 = { by axiom 21 (op_implies_and) R->L }
% 0.19/0.67 fresh22(op_implies_and, true, X, Y)
% 0.19/0.67 = { by axiom 3 (rosser_op_implies_and) }
% 0.19/0.67 fresh22(true, true, X, Y)
% 0.19/0.67 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.67 fresh22(op_or, true, X, Y)
% 0.19/0.67 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.67 fresh22(op_or, op_or, X, Y)
% 0.19/0.67 = { by axiom 17 (op_implies_and) }
% 0.19/0.67 implies(X, Y)
% 0.19/0.67
% 0.19/0.67 Lemma 37: implies(not(X), Y) = or(X, Y).
% 0.19/0.67 Proof:
% 0.19/0.67 implies(not(X), Y)
% 0.19/0.67 = { by lemma 36 R->L }
% 0.19/0.67 not(and(not(X), not(Y)))
% 0.19/0.67 = { by axiom 23 (op_or) R->L }
% 0.19/0.67 fresh20(op_or, true, X, Y)
% 0.19/0.67 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.67 fresh20(op_or, op_or, X, Y)
% 0.19/0.67 = { by axiom 20 (op_or) }
% 0.19/0.67 or(X, Y)
% 0.19/0.67
% 0.19/0.67 Lemma 38: fresh59(X, X, Y, Z) = op_or.
% 0.19/0.67 Proof:
% 0.19/0.67 fresh59(X, X, Y, Z)
% 0.19/0.67 = { by axiom 14 (modus_ponens_2) }
% 0.19/0.67 fresh60(modus_ponens, true, Z)
% 0.19/0.67 = { by axiom 5 (rosser_modus_ponens) }
% 0.19/0.67 fresh60(true, true, Z)
% 0.19/0.67 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.67 fresh60(op_or, true, Z)
% 0.19/0.67 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.67 fresh60(op_or, op_or, Z)
% 0.19/0.67 = { by axiom 11 (modus_ponens_2) }
% 0.19/0.67 true
% 0.19/0.67 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.67 op_or
% 0.19/0.67
% 0.19/0.67 Lemma 39: is_a_theorem(implies(or(X, Y), or(Y, X))) = fresh8(r3, and_1, X, Y).
% 0.19/0.67 Proof:
% 0.19/0.67 is_a_theorem(implies(or(X, Y), or(Y, X)))
% 0.19/0.67 = { by axiom 30 (r3_1) R->L }
% 0.19/0.67 fresh8(r3, true, X, Y)
% 0.19/0.67 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.67 fresh8(r3, op_or, X, Y)
% 0.19/0.67 = { by lemma 35 }
% 0.19/0.67 fresh8(r3, and_1, X, Y)
% 0.19/0.67
% 0.19/0.67 Lemma 40: fresh59(is_a_theorem(implies(X, Y)), op_or, X, Y) = fresh28(is_a_theorem(X), op_or, Y).
% 0.19/0.67 Proof:
% 0.19/0.67 fresh59(is_a_theorem(implies(X, Y)), op_or, X, Y)
% 0.19/0.67 = { by axiom 1 (rosser_op_or) }
% 0.19/0.67 fresh59(is_a_theorem(implies(X, Y)), true, X, Y)
% 0.19/0.67 = { by axiom 28 (modus_ponens_2) }
% 0.19/0.67 fresh28(is_a_theorem(X), true, Y)
% 0.19/0.67 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.68 fresh28(is_a_theorem(X), op_or, Y)
% 0.19/0.68
% 0.19/0.68 Lemma 41: fresh28(is_a_theorem(or(X, Y)), and_1, implies(implies(Y, Z), or(Z, X))) = and_1.
% 0.19/0.68 Proof:
% 0.19/0.68 fresh28(is_a_theorem(or(X, Y)), and_1, implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by lemma 35 R->L }
% 0.19/0.68 fresh28(is_a_theorem(or(X, Y)), op_or, implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by lemma 40 R->L }
% 0.19/0.68 fresh59(is_a_theorem(implies(or(X, Y), implies(implies(Y, Z), or(Z, X)))), op_or, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by lemma 36 R->L }
% 0.19/0.68 fresh59(is_a_theorem(implies(or(X, Y), implies(not(and(Y, not(Z))), or(Z, X)))), op_or, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by lemma 37 }
% 0.19/0.68 fresh59(is_a_theorem(implies(or(X, Y), or(and(Y, not(Z)), or(Z, X)))), op_or, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by lemma 37 R->L }
% 0.19/0.68 fresh59(is_a_theorem(implies(or(X, Y), or(and(Y, not(Z)), implies(not(Z), X)))), op_or, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by lemma 37 R->L }
% 0.19/0.68 fresh59(is_a_theorem(implies(implies(not(X), Y), or(and(Y, not(Z)), implies(not(Z), X)))), op_or, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by lemma 36 R->L }
% 0.19/0.68 fresh59(is_a_theorem(implies(implies(not(X), Y), or(and(Y, not(Z)), not(and(not(Z), not(X)))))), op_or, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by lemma 37 R->L }
% 0.19/0.68 fresh59(is_a_theorem(implies(implies(not(X), Y), implies(not(and(Y, not(Z))), not(and(not(Z), not(X)))))), op_or, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by axiom 32 (kn3_1) R->L }
% 0.19/0.68 fresh59(fresh29(kn3, true, not(X), Y, not(Z)), op_or, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by axiom 8 (rosser_kn3) }
% 0.19/0.68 fresh59(fresh29(true, true, not(X), Y, not(Z)), op_or, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.68 fresh59(fresh29(op_or, true, not(X), Y, not(Z)), op_or, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by lemma 35 }
% 0.19/0.68 fresh59(fresh29(and_1, true, not(X), Y, not(Z)), op_or, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.68 fresh59(fresh29(and_1, op_or, not(X), Y, not(Z)), op_or, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by lemma 35 }
% 0.19/0.68 fresh59(fresh29(and_1, and_1, not(X), Y, not(Z)), op_or, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by axiom 22 (kn3_1) }
% 0.19/0.68 fresh59(true, op_or, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.68 fresh59(op_or, op_or, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by lemma 35 }
% 0.19/0.68 fresh59(and_1, op_or, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by lemma 35 }
% 0.19/0.68 fresh59(and_1, and_1, or(X, Y), implies(implies(Y, Z), or(Z, X)))
% 0.19/0.68 = { by lemma 38 }
% 0.19/0.68 op_or
% 0.19/0.68 = { by lemma 35 }
% 0.19/0.68 and_1
% 0.19/0.68
% 0.19/0.68 Goal 1 (principia_r3): r3 = true.
% 0.19/0.68 Proof:
% 0.19/0.68 r3
% 0.19/0.68 = { by axiom 31 (r3) R->L }
% 0.19/0.68 fresh9(is_a_theorem(implies(or(p3, q3), or(q3, p3))), true)
% 0.19/0.68 = { by lemma 39 }
% 0.19/0.68 fresh9(fresh8(r3, and_1, p3, q3), true)
% 0.19/0.68 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.68 fresh9(fresh8(r3, and_1, p3, q3), op_or)
% 0.19/0.68 = { by lemma 35 }
% 0.19/0.68 fresh9(fresh8(r3, and_1, p3, q3), and_1)
% 0.19/0.68 = { by lemma 39 R->L }
% 0.19/0.68 fresh9(is_a_theorem(implies(or(p3, q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by lemma 37 R->L }
% 0.19/0.68 fresh9(is_a_theorem(implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by axiom 13 (modus_ponens_2) R->L }
% 0.19/0.68 fresh9(fresh28(and_1, and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by lemma 35 R->L }
% 0.19/0.68 fresh9(fresh28(op_or, and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by lemma 38 R->L }
% 0.19/0.68 fresh9(fresh28(fresh59(and_1, and_1, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by lemma 35 R->L }
% 0.19/0.68 fresh9(fresh28(fresh59(and_1, op_or, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by lemma 41 R->L }
% 0.19/0.68 fresh9(fresh28(fresh59(fresh28(is_a_theorem(or(not(p3), and(not(not(p3)), not(not(p3))))), and_1, implies(implies(and(not(not(p3)), not(not(p3))), p3), or(p3, not(p3)))), op_or, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by lemma 37 R->L }
% 0.19/0.68 fresh9(fresh28(fresh59(fresh28(is_a_theorem(implies(not(not(p3)), and(not(not(p3)), not(not(p3))))), and_1, implies(implies(and(not(not(p3)), not(not(p3))), p3), or(p3, not(p3)))), op_or, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by axiom 25 (kn1_1) R->L }
% 0.19/0.68 fresh9(fresh28(fresh59(fresh28(fresh33(kn1, true, not(not(p3))), and_1, implies(implies(and(not(not(p3)), not(not(p3))), p3), or(p3, not(p3)))), op_or, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by axiom 6 (rosser_kn1) }
% 0.19/0.68 fresh9(fresh28(fresh59(fresh28(fresh33(true, true, not(not(p3))), and_1, implies(implies(and(not(not(p3)), not(not(p3))), p3), or(p3, not(p3)))), op_or, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.68 fresh9(fresh28(fresh59(fresh28(fresh33(op_or, true, not(not(p3))), and_1, implies(implies(and(not(not(p3)), not(not(p3))), p3), or(p3, not(p3)))), op_or, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.68 fresh9(fresh28(fresh59(fresh28(fresh33(op_or, op_or, not(not(p3))), and_1, implies(implies(and(not(not(p3)), not(not(p3))), p3), or(p3, not(p3)))), op_or, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by axiom 12 (kn1_1) }
% 0.19/0.68 fresh9(fresh28(fresh59(fresh28(true, and_1, implies(implies(and(not(not(p3)), not(not(p3))), p3), or(p3, not(p3)))), op_or, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.68 fresh9(fresh28(fresh59(fresh28(op_or, and_1, implies(implies(and(not(not(p3)), not(not(p3))), p3), or(p3, not(p3)))), op_or, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by lemma 35 }
% 0.19/0.68 fresh9(fresh28(fresh59(fresh28(and_1, and_1, implies(implies(and(not(not(p3)), not(not(p3))), p3), or(p3, not(p3)))), op_or, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by axiom 13 (modus_ponens_2) }
% 0.19/0.68 fresh9(fresh28(fresh59(is_a_theorem(implies(implies(and(not(not(p3)), not(not(p3))), p3), or(p3, not(p3)))), op_or, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by axiom 18 (op_implies_or) R->L }
% 0.19/0.68 fresh9(fresh28(fresh59(is_a_theorem(implies(fresh21(op_or, op_or, and(not(not(p3)), not(not(p3))), p3), or(p3, not(p3)))), op_or, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by axiom 1 (rosser_op_or) }
% 0.19/0.68 fresh9(fresh28(fresh59(is_a_theorem(implies(fresh21(op_or, true, and(not(not(p3)), not(not(p3))), p3), or(p3, not(p3)))), op_or, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by axiom 1 (rosser_op_or) }
% 0.19/0.68 fresh9(fresh28(fresh59(is_a_theorem(implies(fresh21(true, true, and(not(not(p3)), not(not(p3))), p3), or(p3, not(p3)))), op_or, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by axiom 4 (principia_op_implies_or) R->L }
% 0.19/0.68 fresh9(fresh28(fresh59(is_a_theorem(implies(fresh21(op_implies_or, true, and(not(not(p3)), not(not(p3))), p3), or(p3, not(p3)))), op_or, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by axiom 19 (op_implies_or) }
% 0.19/0.68 fresh9(fresh28(fresh59(is_a_theorem(implies(or(not(and(not(not(p3)), not(not(p3)))), p3), or(p3, not(p3)))), op_or, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by lemma 36 }
% 0.19/0.68 fresh9(fresh28(fresh59(is_a_theorem(implies(or(implies(not(not(p3)), not(p3)), p3), or(p3, not(p3)))), op_or, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by lemma 37 }
% 0.19/0.68 fresh9(fresh28(fresh59(is_a_theorem(implies(or(or(not(p3), not(p3)), p3), or(p3, not(p3)))), op_or, or(or(not(p3), not(p3)), p3), or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by lemma 40 }
% 0.19/0.68 fresh9(fresh28(fresh28(is_a_theorem(or(or(not(p3), not(p3)), p3)), op_or, or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by lemma 35 }
% 0.19/0.68 fresh9(fresh28(fresh28(is_a_theorem(or(or(not(p3), not(p3)), p3)), and_1, or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by lemma 37 R->L }
% 0.19/0.68 fresh9(fresh28(fresh28(is_a_theorem(implies(not(or(not(p3), not(p3))), p3)), and_1, or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by axiom 24 (op_and) R->L }
% 0.19/0.68 fresh9(fresh28(fresh28(is_a_theorem(implies(fresh24(op_and, true, p3, p3), p3)), and_1, or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by axiom 2 (principia_op_and) }
% 0.19/0.68 fresh9(fresh28(fresh28(is_a_theorem(implies(fresh24(true, true, p3, p3), p3)), and_1, or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.68 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.68 fresh9(fresh28(fresh28(is_a_theorem(implies(fresh24(op_or, true, p3, p3), p3)), and_1, or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.70 = { by axiom 1 (rosser_op_or) R->L }
% 0.19/0.70 fresh9(fresh28(fresh28(is_a_theorem(implies(fresh24(op_or, op_or, p3, p3), p3)), and_1, or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.70 = { by axiom 16 (op_and) }
% 0.19/0.70 fresh9(fresh28(fresh28(is_a_theorem(implies(and(p3, p3), p3)), and_1, or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.70 = { by lemma 33 }
% 0.19/0.70 fresh9(fresh28(fresh28(fresh58(and_1, op_or, p3, p3), and_1, or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.70 = { by lemma 34 }
% 0.19/0.70 fresh9(fresh28(fresh28(op_or, and_1, or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.70 = { by lemma 35 }
% 0.19/0.70 fresh9(fresh28(fresh28(and_1, and_1, or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.70 = { by axiom 13 (modus_ponens_2) }
% 0.19/0.70 fresh9(fresh28(is_a_theorem(or(p3, not(p3))), and_1, implies(implies(not(p3), q3), or(q3, p3))), and_1)
% 0.19/0.70 = { by lemma 41 }
% 0.19/0.70 fresh9(and_1, and_1)
% 0.19/0.70 = { by axiom 10 (r3) }
% 0.19/0.70 true
% 0.19/0.71 % SZS output end Proof
% 0.19/0.71
% 0.19/0.71 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------